Isosceles Right Angled Triangle

The three sides cannot be all integers 1 1

Find a right angled triangle with integral sides near to be isosceles such as

Question The difference between two legs is 1 What is the next triangle? Are there sequences of the legs with ratio tending to 1 : 1 ?

a + 1 a a + k The three sides should be where a, k are positive integers and k > 1 a, a + 1 and a + k

Relation between a and k a 2 + (a + 1) 2 = (a + k) 2 a 2 – 2(k – 1)a – (k 2 – 1) = 0 as a > 0

Discriminant 2k(k – 1) is a perfect square For k, k – 1, one is odd, one is even Since k, k – 1 are relatively prime, odd one = x 2, even one = 2y 2 Then x 2 – 2y 2 =  1 2k(k – 1) = 4x 2 y 2

Find the three sides

If k is odd, then k = x 2, then the three sides are: 2y 2 + 2xy, x 2 + 2xy, x 2 + 2xy + 2y 2 If k is even, then k = 2y 2, then the three sides are: x 2 + 2xy, 2y 2 + 2xy, x 2 + 2xy + 2y 2 Both cases have the same form.

Pell’s Equation x 2 – 2y 2 =  1 How to solve it? Making use the expansion of into infinite continued fraction

Continued Fraction

Sequence of the Fractions

{f n } is generated by the sequence {1, 2, 2, 2, …. } where h 1 = 1, h 2 = 3, k 1 = 1, k 2 = 2 h n = h n-2 + 2h n-1, k n = k n-2 + 2k n-1 for n > 2

Solution of Pell’s Equation x 2 – 2y 2 =  1 ( h n, k n ) is a solution of x 2 – 2y 2 = 1 when n is even. (h n, k n ) is a solution of x 2 – 2y 2 = -1 when n is odd. This is the full solution set.

Sequences of the Sides n x = h n y = k n a n =x 2 +2xy b n =2y(x+y) x 2 +2xy+2y

Properties of Sequences The triangle tends to be isosceles